Theorem sibfima 33232

Description: Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)

Hypotheses

Ref	Expression
sitgval.b	⊢ 𝐵 = (Base‘𝑊)
sitgval.j	⊢ 𝐽 = (TopOpen‘𝑊)
sitgval.s	⊢ 𝑆 = (sigaGen‘𝐽)
sitgval.0	⊢ 0 = (0g‘𝑊)
sitgval.x	⊢ · = ( ·𝑠 ‘𝑊)
sitgval.h	⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
sitgval.1	⊢ (𝜑 → 𝑊 ∈ 𝑉)
sitgval.2	⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
sibfmbl.1	⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))

Assertion

Ref	Expression
sibfima	⊢ ((𝜑 ∧ 𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞))

Proof of Theorem sibfima

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sibfmbl.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
2		sitgval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
3		sitgval.j	. . . . 5 ⊢ 𝐽 = (TopOpen‘𝑊)
4		sitgval.s	. . . . 5 ⊢ 𝑆 = (sigaGen‘𝐽)
5		sitgval.0	. . . . 5 ⊢ 0 = (0g‘𝑊)
6		sitgval.x	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
7		sitgval.h	. . . . 5 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
8		sitgval.1	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
9		sitgval.2	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
10	2, 3, 4, 5, 6, 7, 8, 9	issibf 33227	. . . 4 ⊢ (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))))
11	1, 10	mpbid 231	. . 3 ⊢ (𝜑 → (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))
12	11	simp3d 1144	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))
13		sneq 4633	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
14	13	imaeq2d 6050	. . . . 5 ⊢ (𝑥 = 𝐴 → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {𝐴}))
15	14	fveq2d 6883	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑀‘(◡𝐹 “ {𝑥})) = (𝑀‘(◡𝐹 “ {𝐴})))
16	15	eleq1d 2818	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞)))
17	16	rspcv 3606	. 2 ⊢ (𝐴 ∈ (ran 𝐹 ∖ { 0 }) → (∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞)))
18	12, 17	mpan9 507	1 ⊢ ((𝜑 ∧ 𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ∖ cdif 3942  {csn 4623  ∪ cuni 4902  ^◡ccnv 5669  dom cdm 5670  ran crn 5671   “ cima 5673  ‘cfv 6533  (class class class)co 7394  Fincfn 8924  0cc0 11094  +∞cpnf 11229  [,)cico 13310  Basecbs 17128  Scalarcsca 17184   ·_𝑠 cvsca 17185  TopOpenctopn 17351  0_gc0g 17369  ℝHomcrrh 32868  sigaGencsigagen 33031  measurescmeas 33088  MblFnMcmbfm 33142  sitgcsitg 33223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-sitg 33224

This theorem is referenced by:  sibfinima  33233  sitgfval  33235  sitgclg  33236